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Mirrors > Home > MPE Home > Th. List > nss | Structured version Visualization version GIF version |
Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
nss | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exanali 1856 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | df-ss 3979 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | xchbinxr 335 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ¬ 𝐴 ⊆ 𝐵) |
4 | 3 | bicomi 224 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 ∃wex 1775 ∈ wcel 2105 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-ss 3979 |
This theorem is referenced by: grur1 10857 psslinpr 11068 reclem2pr 11085 mreexexlem2d 17689 prmcyg 19926 filconn 23906 alexsubALTlem4 24073 wilthlem2 27126 shne0i 31476 erdszelem10 35184 fundmpss 35747 ntrneineine1lem 44073 nssrex 45025 nssd 45044 nsssmfmbf 46734 |
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